Associated to any simplicial graph there is a right-angled Coxeter group. Invariants of the Coxeter group such as its growth series or its weighted L^2 Betti numbers can be computed from the graph's clique complex (i.e., its flag complex).
The systolic inequality says that if we take any metric on an n-dimensional torus with volume 1, then we can find a non-contractible curve in the torus with length at most C(n). A remarkable feature of the inequality is how general it is: it holds for all metrics.
In a sequence of recent papers, Sudan and coauthors have investigated the relation between testability of properties of Boolean functions and the invariance of the properties with respect to transformations of the domain. Linear-invariance is arguably the most common such symmetry for natural properties of Boolean functions on the hypercube. Hence, it is an important goal to find necessary and sufficient conditions for testability of linear-invariant properties.
The early work of Condorcet in the eighteenth century, and that of Arrow and others in the twentieth century, revealed the complex and interesting mathematical problems that arise in the theory of social choice. In this lecture, Noga Alon, Visiting Professor in the School of Mathematics, explains how the simple process of voting leads to strikingly counter-intuitive paradoxes, focusing on several recent intriguing examples.
Finding the longest increasing subsequence (LIS) is a classic algorithmic problem. Simple $O(n log n)$ algorithms, based on dynamic programming, are known for solving this problem exactly on arrays of length $n$.